(In reply to Is my reasoning valid?
I don't think your "proof' is quite valid either, but I certainly think you are on to something.
The expression is not infinite for x <=2, and by substitution we can see easily enough that f(2) = 2
But I don't see offhand how to use induction to get f(3) = 3,
because (f(x)^2 - x - 2)/(x-2) is undefined when x = 2
If we could prove that f(3) = 3, then it follows that f(x) = x.